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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 13 — Dec. 17, 2001
  • pp: 813–820
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Ring structures in microstructured polymer optical fibres

Alexander Argyros, Ian M. Bassett, Martijn A. van Eijkelenborg, Maryanne C.J. Large, Joseph Zagari, Nicolae A.P. Nicorovici, Ross C. McPhedran, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 9, Issue 13, pp. 813-820 (2001)
http://dx.doi.org/10.1364/OE.9.000813


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Abstract

Recent developments in polymer microstructured optical fibres allow for the realisation of microstructures in fibres that would be problematic to fabricate using glass-based capillary stacking. We present one class of such structures, where the holes lie on circular rings. A fibre of this type is fabricated and shown to be single moded for relatively long lengths of fibre, whereas shorter lengths are multimoded. An average index model for these fibres is developed. Comparison of its predictions to the calculated properties of the exact structure indicates that the ring structures emulate homogeneous rings of lower refractive index resulting in the ring structured fibres behaving approximately as cylindrically layered fibres.

© Optical Society of America

1. Introduction

Recent developments in the fabrication [1

1. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

,2

2. H. Kubota, K. Suzuki, S. Kawanishi, M. Kakazawa, M. Tanaka, and M. Fujita, “Low-loss, 2 km-long photonic crystal fibre with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band,” Postdeadline paper CPD3, Conference on Lasers and Electro-Optics CLEO 2001, (Optical Society of America, Washington, D.C., 2001)

], and modelling [3

3. E. Silvestre, M.V. Andres, and P. Andres, “Biorthonormal-basis method for the vector description of optical fiber modes,” J. Lightwave Technol 16, 923–928 (1998). [CrossRef]

6

6. T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 1660–1662 (2001). [CrossRef]

], of (silica) microstructured optical fibres (MOFs) have been motivated by their interesting properties which make MOFs suitable for a wide range of applications. The use of polymer materials instead of silica for the fabrication of these fibres allows for an even wider range of optical properties to be obtained [7

7. M.A. van Eijkelenborg, M.C.J. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. de Sterke, and N.A.P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express 9, 319–327 (2001), http://www.opticsexpress.org/oearchive/source/35051.htm. [CrossRef] [PubMed]

]. This advantage arises from the variety of polymers that are available and from the ease with which these are processed by e.g. drilling, extrusion, casting, polymerisation inside a mould and injection moulding. As a result, microstructured polymer optical fibres (MPOFs) can be fabricated with almost arbitrary hole structures, not restricted to the hexagonal or square arrangements of holes that are obtained with the capillary stacking technique used to make silica MOFs. The drawing process of MPOF seems to be very robust; differences between the structure in the preform and the fibre were found to be minimal [7

7. M.A. van Eijkelenborg, M.C.J. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. de Sterke, and N.A.P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express 9, 319–327 (2001), http://www.opticsexpress.org/oearchive/source/35051.htm. [CrossRef] [PubMed]

]. In contrast the structure in silica MOFs often suffers from large deformations due to the strong effects of surface tension in the draw process.

2. Fabricated ring structured fibres

We start by describing the ring structured MPOFs that we have fabricated. The PMMA we used was of relatively low optical quality; the material losses (absorption and scattering) were measured to be about 4 dB/m, as determined in a measurement with polished rods of different lengths. Better quality material is intended for future work. The preform was formed by stacking 1 mm thick structured cane with capillaries inside a tube; this method allows the study of many structured sections with a single fibre. The structured sections of the cane consisted of three equally spaced rings of holes with the same air-filling fraction in each ring. A cross section of the neck-down region of a preform is shown in Fig. 1. In this image several microstructured sections can be seen with three equally spaced rings. The core in the lower left corner is surrounded by relatively large holes, which result in large index contrasts between the high and low index layers, whereas a lower index contrast is expected for the core closer to the centre which is surrounded by smaller holes.

The preform shown in Fig. 1 was drawn down to fibre of about 260 µm diameter on a draw tower with feed speed of 4 mm/min and draw speed of no more than 5 m/min at a temperature of approximately 175 °C. An electron microscope image of the cross section of the fibre can be seen in Fig. 2, which shows that the dimensions of the holes in the final structure are of the order of a visible wavelength. Optical transmission experiments were carried out on this fibre using HeNe laser light and a white light source. Many different cores were found to guide both white and HeNe light over distances of a few metres.

Fig. 1. Optical micrograph of the cross section of the preform neck-down region. The image width corresponds to approximately 1.5 mm. Several ring structured sections can be seen with different sized holes. The structure on which the optical experiments were conducted and which was modelled is that in the lower left corner with the larger holes.
Fig. 2. Electron micrograph of a cross section of the microstructured polymer optical fibre.

Considering only the core that resulted from the structured section in the lower left corner of Fig. 1, fibre sections of 25 cm in length and longer were experimentally found to be single moded by performing the usual tests for single modedness as reported in [1

1. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

,7

7. M.A. van Eijkelenborg, M.C.J. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. de Sterke, and N.A.P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express 9, 319–327 (2001), http://www.opticsexpress.org/oearchive/source/35051.htm. [CrossRef] [PubMed]

], including launching the HeNe directly into the fibre with a microscope objective and observing that the mode profile is insensitive to bending of the fibre and changing of the launch conditions. However, when the fibre was cut back to 5 cm, the same core was found to support more than one mode, as indicated by the observation of interference between different modes in the near field pattern, which changed as a function of the launch conditions (by translating the microscope objective that focused the laser beam onto the core). This behavior is explained in Section 4, and illustrates the meaning of effective single modedness of microstructured fibres in which the fibre length is an important parameter [6

6. T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 1660–1662 (2001). [CrossRef]

,7

7. M.A. van Eijkelenborg, M.C.J. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. de Sterke, and N.A.P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express 9, 319–327 (2001), http://www.opticsexpress.org/oearchive/source/35051.htm. [CrossRef] [PubMed]

].

3. Microstructures and ring structures

Features in a structure that are comparable to or smaller than a wavelength are, in general, not expected be resolved clearly by the light. If these features are inclusions of a material with some refractive index imbedded in a matrix of a different refractive index, the composite material can approximately be replaced by a homogenised or effective medium. The refractive index n av of the homogenised material is an average of the refractive indices of the inclusions n inc and the matrix n matrix, weighted by the filling fraction f of the inclusions, which in general satisfies [13

13. G.W. Milton, The theory of composites, (Cambridge University Press, London, 2001).

]

(fninc2+1fnmatrix2)12navfninc2+(1f)nmatrix2.
(1)

The light in the fibre propagates in a direction almost parallel to that of the air channels. Therefore the relevant quantity that determines whether the holes are resolved, i.e. whether averaging is justified, is not the wavelength itself. It is the scaled reciprocal of the radial wavenumber 2π/kr (with kr the component of wavevector in the radial direction, perpendicular to the holes), which is much larger than the wavelength. Based on this argument, nearly arbitrary index profiles can be constructed in microstructured fibres. Applying the averaging to the ring structured fibres means that each ring of holes will effectively behave like a homogeneous ring of refractive index lower than that of the polymer (see Fig. 3).

Fig. 3. (a) A circular ring of holes is expected to behave like a circular layer of lower refractive index (b), the corresponding refractive index profile is shown in (c). Several equally spaced rings of holes of equal air-filling fraction will result in an index profile such as that shown in (d), i.e. a Bragg fibre.

An advantage of using ring structures, apart from the flexibility in the structure, is the large range of refractive indices possible. The maximum air-filling fraction obtainable using a ring of holes occurs when the holes are just touching, which corresponds to a value of 0.79 for the entire ring. Using Eq. (1) for our fibre material (PMMA) with a refractive index of 1.5 in the visible, this corresponds to an index difference of about 0.4. This index contrast is much larger than index differences obtainable in silica fibres by processes such as modified chemical vapour deposition (MCVD).

The average index of each layer can be adjusted by varying the size, shape and spacing of the holes in each ring. The shape of the index profile versus radius associated with each ring depends on the hole size and shape. For example, a ring of circular holes will result in a smooth graded index profile. Rings of annular shaped holes or many closely spaced rings of small holes will result in a step-index profile for each ring. Obviously the latter can be used to achieve much higher air-filling fractions and hence higher index contrasts.

The reason large index contrasts are desirable is the possibility of fabricating Bragg fibres using MPOF techniques. A structure with equally spaced concentric rings of holes and the same air-filling fraction in each ring will, if the size of the structure meets the conditions discussed, emulate a refractive index profile of a Bragg fibre (see Fig. 3). The periodicity in the radial direction will give rise to Bragg reflection confining the light to the core and the large index contrasts will increase the strength of the reflection and therefore minimise the number or rings required to achieve adequate confinement. Such large and complex variations in refractive index are difficult to obtain by conventional means in silica and polymer fibres, which are limited by e.g. maximum dopant concentrations. Very high index contrast cylindrical Bragg structures, with index contrasts of 3, have been made from a combination of different materials [10

10. Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041(1999). [CrossRef]

], but they are limited in length by the method of fabrication used. Therefore, ring structured MPOFs may open the way to making long lengths of high index contrast Bragg fibre.

4. The average index model

The average index profile for a particular hole structure is found by averaging the refractive index over 360° for fixed values of the radius. The upper and lower limits of Eq. (1) were used to calculate the average index for each radius as well as just the arithmetic mean of the refractive index, which is given by

nav=fninc+(1f)nmatrix,
(2)

and is shown in Fig. 4. Once the average index profile is calculated the guided modes can be found using Chew’s method [14

14. W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York1990).

]; an exact method for finding the modes of circularly symmetric fibres.

Chew’s method solves for the modes of a fibre by solving the vector wave equation in each layer of the fibre and combining the solutions from each layer. The solutions are combined using 2×2 matrices derived from the boundary conditions and an eigenvalue condition (corresponding to the elimination of an unphysical singularity of the fields at the origin) is imposed to find the modes. Modifications were made to the method to allow for evanescent regions within the fibre and the calculation of confinement losses of leaky modes. The confinement losses are calculated by comparing the radial and longitudinal components of the Poynting vector for each mode.

In addition to the average index calculations, we modeled the ring structured MPOF using a recently developed multipole method [6

6. T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 1660–1662 (2001). [CrossRef]

]. Compared with other modeling techniques, it has advantages in terms of accuracy and the ability to calculate confinement losses of microstructured fibres with circular holes. The results of the multipole method allow us to compare the approximate average index model to the calculations for the exact holey structure.

The structure of the fibre that was modelled (the core on which the optical tests were conducted) consisted of holes of 0.7 µm radius in the first two rings and 0.6 µm holes in the third ring. The radii of the three rings were 2.9, 5.8 and 8.5 µm — the 0.2 µm difference in radius between the ‘perfect structure’, with the third ring at 8.7 µm radius, and the real fibre dimensions is caused by the drawing process. The hole size and spacing is of the order of 1 µm, which is sufficient for the average index model, since it is much smaller than the scaled reciprocal of the radial wavenumber as mentioned before. The average index calculated for the structure is presented in Fig. 4. Both calculation methods use a lossless matrix of index n=1.4897 for PMMA at 632.8 nm, corresponding to the wavelength used in the experiments.

Figure 4. The average index profile calculated for the structure described in the text. The averaging method used was to take the arithmetic mean of the refractive index over 360° for fixed values of the radius. This gives a value for the average between the two limits of Eq. (1). The average index profile is constructed out of 40 layers.

The results of both modelling techniques for this fibre are given in Table 1. In the case of the multipole method, values of the n eff of the modes, as well as the confinement losses are presented for the structure with only the inner ring of holes included and with the inner two rings included. The modeling was limited to the inner two rings of holes due to the large amount of time that would be required for the calculations of the full three ring structure. There was a large decrease in confinement loss when the second ring was included in the calculations. However, the inclusion of the second ring of holes had a negligible effect on the n eff values of the modes. The effect of the third ring on n eff would be even smaller given the lower mode amplitudes at the location of the third ring.

For the average index calculations n av was calculated using both limits in Eq. (1) as well as the arithmetic mean of the refractive index. Differences in the values of n eff of the modes for the three cases were found to be of the order of 10-4 and always smaller than the difference between the average index model n eff and the values of n eff calculated using the multipole method. Therefore only the results for the arithmetic mean are presented in Table 1. The modes predicted by the two models when ordered by effective index firstly agreed in class (where modes in the same class have the same dependence m on azimuthal angle, using notation as in [14

14. W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York1990).

]) and secondly in the values of the effective indices (with differences of order 10-4 to 10-3 for the higher-order modes). This is a strong indication that a microstructured fibre with holes positioned on concentric rings shows similar behaviour to a cylindrically layered structure.

Table 1. Mode class and effective indices of the first five modes as calculated by the multipole method and the average index model, using the arithmetic mean of the refractive index.

table-icon
View This Table

It must be noted that the classification of the modes is only approximate for the results of the multipole case. For example, modes classified as TE have a non-zero longitudinal electric field component, though it is much smaller than the longitudinal magnetic field and is only be non-zero near the holes. It can also be seen in Table 1 that the effective indices of the modes predicted by the average index model are all lower than those calculated using the multipole method. This occurs because in the microstructured fibre, the light concentrates between the holes of the first ring, rather than penetrating into the air, which results in the light “seeing” more polymer than air. Therefore, effectively the index of the first ring is slightly higher than that calculated by averaging, resulting in higher effective indices for the modes. It was found that if the average index of the first ring was raised so that the neff of the fundamental mode as calculated by the average index model matched that predicted by the multipole method, the differences between the neff values of the higher-order modes reduced by an order of magnitude.

The confinement losses of the modes were also calculated with both the average index model and the multipole method (the latter again limited to two rings of holes). The results for the multipole method are included in Table 1. Unsurprisingly, the losses calculated with the two methods do not agree. The average index model predicts confinement losses many orders of magnitude lower than the multipole method. This is because in the microstructured fibre the main loss mechanism is the leakage of light through the high index paths in between the holes. This leakage route does not exist in the angularly averaged index fibre, where all holes are replaced by closed rings. The loss calculated by the average index model is that which would arise from tunnelling, and hence is much lower.

5. Conclusion

We have shown that it is possible to fabricate microstructured optical fibres with structures other than what can be achieved by the conventional capillary stacking methods used for glass microstructured fibres. In particular, we have fabricated a microstructured polymer fibre in which the structure consists of circular rings of holes. This fibre was found to be single moded for relatively long lengths, but multimoded for short lengths — a property observed here for the first time and unique to microstructured fibres. These experimental results and the theoretical arguments based on the confinement losses as calculated by the multipole method form the basis of a definition of single-modedness in microstructured fibres: a microstructured fibre of a certain length is in effect single-moded when all modes except the fundamental have a confinement loss that is too large to observe transmission in any modes except the fundamental mode.

The rings of holes in the fibre are shown to approximate rings of homogeneous refractive index, determined by the air-filling fraction in each ring. Qualitative theoretical results for this type of structure were obtained using an angular effective index model. Comparing these results to the multipole method for the actual fibre microstructure has shown that the average index model offers a fast qualitative alternative for modelling ring microstructured fibres. The high index contrasts possible in ring structured fibres offer a method for the fabrication of long lengths of high index contrast Bragg fibres. Further investigation of the loss mechanisms is intended; the use of a low loss polymer will allow the effects of hole surface roughness and deformations in structure to be studied.

Acknowledgements

The authors would like to acknowledge the assistance of Barry Reed, Nader Issa and Geoff Henry. The electron micrographs were provided by the Electron Microscope Unit of the University of Sydney. The project was funded by the Australian Photonics Cooperative Research Centre and Redfern Polymer Optics Pty. Ltd.

References and links

1.

T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

2.

H. Kubota, K. Suzuki, S. Kawanishi, M. Kakazawa, M. Tanaka, and M. Fujita, “Low-loss, 2 km-long photonic crystal fibre with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band,” Postdeadline paper CPD3, Conference on Lasers and Electro-Optics CLEO 2001, (Optical Society of America, Washington, D.C., 2001)

3.

E. Silvestre, M.V. Andres, and P. Andres, “Biorthonormal-basis method for the vector description of optical fiber modes,” J. Lightwave Technol 16, 923–928 (1998). [CrossRef]

4.

D. Mogilevtsev, T.A. Birks, and P.St.J. Russell, “Localised function method for modelling defect modes in 2-D photonic crystals,” J. Lightwave Technol. 17, 2078–2081 (2000). [CrossRef]

5.

T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Holey optical fibres: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

6.

T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. 26, 1660–1662 (2001). [CrossRef]

7.

M.A. van Eijkelenborg, M.C.J. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. de Sterke, and N.A.P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express 9, 319–327 (2001), http://www.opticsexpress.org/oearchive/source/35051.htm. [CrossRef] [PubMed]

8.

J. Xu, J. Song, C Li, and K. Ueda, “Cylindrically symmetric hollow fiber,” Optics Commun. 182, 343–348 (2000). [CrossRef]

9.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978). [CrossRef]

10.

Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041(1999). [CrossRef]

11.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero dispersion at short wavelengths,” Electron. Lett. 36, 514–515 (2000) [CrossRef]

12.

M. Ibanescu, Y. Fink, S. Fan, E.L. Thomas, and J.D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000) [CrossRef] [PubMed]

13.

G.W. Milton, The theory of composites, (Cambridge University Press, London, 2001).

14.

W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York1990).

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(160.5470) Materials : Polymers
(230.4000) Optical devices : Microstructure fabrication

ToC Category:
Research Papers

History
Original Manuscript: December 5, 2001
Published: December 17, 2001

Citation
Alexander Argyros, Ian Bassett, Martijn van Eijkelenborg, Maryanne Large, Joseph Zagari, Nicolae A. Nicorovici, Ross McPhedran, and C. Martijn de Sterke, "Ring structures in microstructured polymer optical fibres," Opt. Express 9, 813-820 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-13-813


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References

  1. T.A. Birks, J.C. Knight, P.St.J. Russell, "Endlessly single-mode photonic crystal fibre," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  2. H. Kubota, K. Suzuki, S. Kawanishi, M. Kakazawa, M. Tanaka, M. Fujita, "Low-loss, 2 km-long photonic crystal fibre with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band," Postdeadline paper CPD3, Conference on Lasers and Electro-Optics CLEO 2001, (Optical Society of America, Washington, D.C., 2001)
  3. E. Silvestre, M.V. Andres, P. Andres, "Biorthonormal-basis method for the vector description of optical fiber modes," J. Lightwave Technol 16, 923-928 (1998). [CrossRef]
  4. D. Mogilevtsev, T.A. Birks, P.St.J. Russell, "Localised function method for modelling defect modes in 2-D photonic crystals," J. Lightwave Technol. 17, 2078-2081 (2000). [CrossRef]
  5. T.M. Monro, D.J. Richardson, N.G.R. Broderick, P.J. Bennett, "Holey optical fibres: an efficient modal model," J. Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
  6. T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, M.J. Steel, "Confinement losses in microstructured optical fibres," Opt. Lett. 26, 1660-1662 (2001). [CrossRef]
  7. M.A. van Eijkelenborg, M.C.J. Large, A. Argyros, J. Zagari, S.Manos, N. Issa, I. Bassett, S. Fleming, R.C. McPhedran, C.M. de Sterke, N.A.P. Nicorovici, "Microstructured polymer optical fibre," Opt. Express 9, 319-327 (2001), http://www.opticsexpress.org/oearchive/source/35051.htm. [CrossRef] [PubMed]
  8. J. Xu, J. Song, C Li, K. Ueda, "Cylindrically symmetric hollow fiber," Opt. Commun. 182, 343-348 (2000). [CrossRef]
  9. P. Yeh, A. Yariv, E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
  10. Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, E.L. Thomas, "Guiding optical light in air using an all-dielectric structure," J. Lightwave Technol. 17, 2039-2041(1999). [CrossRef]
  11. F. Brechet, P. Roy, J. Marcou, D. Pagnoux, "Singlemode propagation into depressed-core-index photonicbandgap fibre designed for zero dispersion at short wavelengths," Electron. Lett. 36, 514-515 (2000) [CrossRef]
  12. M. Ibanescu, Y. Fink, S. Fan, E.L. Thomas, J.D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000) [CrossRef] [PubMed]
  13. G.W. Milton, The theory of composites, (Cambridge University Press, London, 2001).
  14. W.C. Chew, Waves and fields in inhomogeneous media, Chapter 3 (Van Nostrand Reinhold, New York 1990).

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